Question

A frame 2 inches wide is placed around a rectangular picture with dimensions 8 inches by 12 inches. What is the area of the frame, in square inches? $$\begin{array}{l}44\\ 96\\ 128\\ 144\\ 168\end{array}$$

Short answer

The area of the frame is 96 square inches.

Step-by-step solution

Determine the dimensions of the framed picture

Add the width of the frame to each side of the picture. Since the frame is 2 inches wide and is placed around the entire picture, add 4 inches to both the width and the height of the picture.

Calculate the total dimensions

The original dimensions of the picture are 8 inches by 12 inches. Adding the frame (2 inches on each side), the new dimensions are: Width: 8 inches + 4 inches = 12 inches Height: 12 inches + 4 inches = 16 inches

Find the area of the framed picture

Calculate the area of the entire framed picture using the total dimensions. Area = width × height = 12 inches × 16 inches = 192 square inches

Calculate the area of the original picture

The original area of the picture is given by the original dimensions. Area = width × height = 8 inches × 12 inches = 96 square inches

Find the area of the frame

The area of the frame is the area of the entire framed picture minus the area of the original picture. Area of the frame = 192 square inches - 96 square inches = 96 square inches

Key concepts

Area Calculation

Understanding how to calculate the area is crucial for solving many geometry problems. Area represents the amount of space within a shape. For rectangles, this is done by multiplying the length by the width. In this exercise, we first calculated the area of the entire framed picture and then the area of the original picture. By subtracting the two, we found the area of the frame. Follow this method whenever asked to find the area of a portion within another shape.

Rectangular Dimensions

An essential part of geometry involves understanding the dimensions of a rectangle. A rectangle has two lengths, usually referred to as length and width. In our exercise, the rectangular picture initially had dimensions of 8 inches by 12 inches. Adding a frame around the picture increased these dimensions. You need to add twice the frame width to both the length and the width, as the frame goes around the entire rectangle. When these adjusted dimensions were determined, the problem became more straightforward.

Geometry Problem

Geometry often involves solving problems that require visualizing changes to shapes. This problem required understanding the effect of adding a frame to a rectangle. Conceptualizing this geometry problem helps to see that the width and height increase by twice the width of the frame, as the frame is added to all sides. Geometry problems become easier when breaking them down and addressing each part step-by-step.

Math Reasoning

Math reasoning is the ability to think logically about a math problem and to solve it systematically. Here, it involved several steps: understanding the effect of the frame on the dimensions; calculating the new dimensions' area; and finally, subtracting the original area from the new area to find the area of the frame. When approaching such problems, logical progression and clear step sequencings, like those shown in the solution steps, are the keys to accurate and efficient problem-solving.